Questions tagged [line-bundles]

For questions about line bundles, that is vector bundles of rank $1$, over topological spaces.

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A line bundle $L$ can be assumed to be ample (after eventually passing to its dual)

I'm reading Huybrechts' Lectures on K3 Surfaces and I got stuck reading example 2.3.9, which shows that any K3 surface $X$ with $\operatorname{Pic}(X)=\mathbb{Z}\cdot L$ and such that $(L)^2=4$ can ...
WindUpBird's user avatar
1 vote
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Construction of a connection 1-form

Let $E \to (M,g)$ be a vector bundle over $M$ a compact riemannian manifold and consider the structural group $G$ of bundle. We can define a linear connection over $E$ as a map $\nabla: \Gamma(E) \to \...
Bri3.1's user avatar
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4 votes
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Local trivialization of $\mathcal O(-1)$, proposition 2.2.6, complex geometry by Huybrechts

I was reading Complex Geometry by Daniel Huybrechts. On page 68, section 2.2 we have a proposition of holomorphic line bundle over $\mathbb P^n$, Proposition 2.2.6: The projection $\pi:\mathcal O(-1)\...
N00BMaster's user avatar
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The tensor product of a line bundle with its dual $L\otimes L^*$ is isomorphic to the trivial line bundle

Let $L$ be a holomorphic line bundle and $L^*$ be the dual holomorphic line bundle. So I believe what we want to show is that the following diagram commutes (plus some condition on the restriction ...
領域展開's user avatar
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4 votes
1 answer
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Does the direct sum of two real line bundles admit a non-vanishing section?

Let $M$ be a smooth manifold, and $L$ be a smooth real (non-trivial) line bundles over $M$, is it then true that $L\oplus L$ admits a non vanishing section? Intuitively, I feel like the answer should ...
Chris's user avatar
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1 vote
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Application of the seesaw principle

The seesaw principle says in general the following. If $X,T$ are varieties with $X$ complete, and $\mathcal{L}$ a line bundle on $X\times T$, then $$ T_{1} = \{t\in T: \mathcal{L}_{X\times\{t\}} \...
user758193's user avatar
2 votes
1 answer
91 views

Normal bundle of transverse intersection of two irreducible components

Let $X$ be an equidimensional reduced scheme of finite type over an algebraically closed field $k$. Assume that $X$ has two irreducible components $X_1$ and $X_2$. Assume also that $X_1$ and $X_2$ are ...
Suzet's user avatar
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2 votes
1 answer
79 views

When is a nef line bundle big

Suppose $M^n$ is a smooth projective variety. A line bundle $L$ on $M$ is nef (numerically effective) if on any complete curve $C$ in $M$, $L$ has positive degree, i.e. $$ L\cdot C=\int_{C}R_h\geq 0. $...
eulershi's user avatar
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Tangent space of $RP^n$ vs the orthogonal complement to the line bundle $\gamma_n^1$

I am reading Milnor's lectures on characteristic classes. He defines the canonical line bundle $\gamma_n^1$ as the set of points $(\pm x, v) \in \mathbb{R}P^n \times \mathbb R^{n+1}$, where $v = tx$ ...
JZweifler's user avatar
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Mapping degree 2 line bundle to a degree 1 line bundle on an elliptic curve

Suppose I have a degree 2 line bundle $L$ on an elliptic curve $E$. Then we may write $L = \mathcal{O}(p+q)$ for some points $p,q \in E$. If we let $\oplus$ denote the addition map on the elliptic ...
Slim Shady's user avatar
1 vote
0 answers
31 views

A question regarding linearized line bundles on a variety with a group action

Let $G$ be an affine algebraic group, and let $X$ be a smooth variety with a $G$-action $\sigma:G\times X\rightarrow X$. Let $\pi: G\times X \rightarrow X$ denote the second projection. A $G$-...
Hajime_Saito's user avatar
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Tensor product of a very ample line bundle with a torsion line bundle

Is there an example of smooth projective surface $S$ having a very ample line bundle $\mathcal{L}$ and a $2$-torsion line bundle $\mathcal{T}$ such that $\mathcal{L}\otimes(\mathcal{T}^{-1})$ has no ...
MaryMoon's user avatar
1 vote
1 answer
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2.3.A Zariski's construction (PAG1 - R. Lazarsfeld), big and nef divisor which is not finitely generated

I'm reading Lazarfeld's book «Positivity in Algebraic Geometry I» and I'm stuck on the construction of a big and nef divisor on a variety $X$ such that its canonical ring/algebra $R(X,D) = \bigoplus_{...
NaNoS's user avatar
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1 answer
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Example 2.13 in Wells "Differential Analysis on Complex Manifolds" Conclusion

I'm currently working through Raymond O. Wells' "Differential Analysis on Complex Manifolds" and I'm confused by example 2.13 in chapter 1. In this example he is computing the global ...
geometric_20's user avatar
4 votes
1 answer
140 views

Pullback of a very ample line bundle under an étale covering

I would like to find an example of very ample line bundle on a smooth projective variety whose pull-back under an étale covering is non-very ample. More precisely: Is there an example of very ample ...
バレリオ's user avatar
0 votes
1 answer
159 views

The canonical sheaf of hypersurface on $\mathbb{P}^n$

I want to prove the following result Let $X\subset\mathbb{P}^n$ be a smooth projective variety defined over $k.$ Let $Y= V(f)$ be a smooth subvariety of $X$ defined by a homogeneous polynomial $f$ of ...
ym2333's user avatar
  • 41
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0 answers
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Constructing unique $2:1$ covering of a non-orientable, connected, smooth manifold

Let $M$ be a non-orientable, connected, smooth manifold with $\text{dim }M=n$. I'm trying to fill in the ideas of a construction of a unique $2:1$ covering $\tilde{M}$ of $M$. I've pieced together ...
Rough_Manifolds's user avatar
-2 votes
1 answer
137 views

Section of a positive degree vector bundle.

Suppose $s\in E$ is a section of a positive degree vector bundle on an algebraic variety $X$. Then naturally, I can always obtain a short exact sequence $0\to O_X\to E\to F\to 0$ defiend by $s$ and $F$...
Shrugs's user avatar
  • 1,458
2 votes
2 answers
64 views

Understanding proof that $E$ and $E^\ast$ are isomorphic rank $1$ bundles.

I would like to prove the following: Proposition. Let $E$ be any real line bundle over $M$. Then $E$ and $E^\ast$ are isomorphic line bundles. I have sketched what I believe works, but am having ...
Rough_Manifolds's user avatar
0 votes
1 answer
199 views

Non zero global section of line bundle and its dual $\implies \mathcal L\cong O_X$

Let $X$ be a projective integral scheme over $K=\bar{K}$. Suppose that $\Gamma(X,\mathcal L)\neq 0$, $\Gamma(X,\mathcal L^{*})\neq 0$. We want to show that $\mathcal L\cong \mathcal O_X$. There is an ...
raisinsec's user avatar
  • 439
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1 answer
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Determining if $\mathbb P^2_K\to \mathbb P^4_K$ is a closed immersion.

Let $K=\bar{K}$ a field, consider the line bundle $\mathcal O_{\mathbb P^2_K}(2)$ and global sections $s_0=x_0^2,s_1=x_1^2,s_2=x_0x_1,s_3=x_0x_2,s_4=x_2^2$ and $\phi:\mathbb P^2_K\to \mathbb P^4_K$ ...
raisinsec's user avatar
  • 439
1 vote
1 answer
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Extension of line bundle on regular deformation

Let $R$ be a DVR and $\pi : S \rightarrow \text{Spec}(R)$ a regular smoothing of a nodal curve $C$ (with regular components). Given a line bundle $L$ on the (regular) generic fiber $\pi^{-1}(\eta)=S_\...
IMP's user avatar
  • 97
1 vote
1 answer
103 views

Base locus of the linear system $|H\otimes\mathfrak{m}_x^2|$ for a very ample divisor

Let $X$ be a smooth complex projective variety of dimension $n\geq1$ and let $H$ be a very ample line bundle on $X.$ Suppose $X\ncong\mathbb{P}^n.$ Fix a closed point $x\in X$ and denote by $\mathfrak{...
MaryMoon's user avatar
2 votes
0 answers
50 views

Integral first Chern class of the line bundle associated with a character

Let $X$ be a connected complex projective manifold, $\chi:\pi_1(X)\to S^1$ be a character of the fundamental group of $X$. Then $\chi$ induces a local system $\mathcal{L}_{\chi}$ of rank $1$ on $X$ ...
Doug's user avatar
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2 votes
1 answer
197 views

Computing the Canonical bundle $K_{\mathbb{P}^n} \cong \mathcal{O}(-n-1)$ of projective space from the Euler sequence

I am reading Huybrechts's Complex Geometry, p.92~p.92 and stuck at showing that $K_{\mathbb{P}^n} \cong \mathcal{O}(-n-1)$ from the Euler sequence. Here, $K_{\mathbb{P}^n}$ is the canonical bundle of ...
Plantation's user avatar
  • 2,417
2 votes
1 answer
62 views

Is the set where $B\otimes\mathfrak{m}_y$ is globally generated, with $B$ a globally generated ample line bundle, non-empty?

Let $X$ be a complex projective variety, and let $B$ be an ample and globally generated line bundle on $X.$ By Example 1.2.9 in Lazarsfeld's book, the set $U$ of points $y\in X$ such that $B\otimes\...
MaryMoon's user avatar
0 votes
0 answers
46 views

Show that the map $\varphi:K \to L, \ \ (x,t) \mapsto (x,t^2)$ is a proper holomorphic map.

Let $X$ be a complex manifold and $L$ be a holomorphic line bundle on $X$. We assume that there exist a holomorphic line bundle $K$ on $X$ and an isomorphism $$K^{\otimes 2} \cong L.$$ Show that the ...
Danlo's user avatar
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Non-zero homogeneous polynomials can be considered as sections of $\mathcal{O}(k)$

This is a passage from a complex geometry lecture notes Any homogeneous polynomial $s \in \mathbb{C}[z_0,...,z_n]_k$ with degree $k$ defines a linear map $(\mathbb{C}^{n+1})^{\otimes k} \to \mathbb{C}...
Danlo's user avatar
  • 735
1 vote
1 answer
96 views

Geometry of homogeneous spaces $G/T$

Let us work over $\mathbb C$. It is well known that if $G$ is a (connected) complex reductive group and $B$ a Borel subgroup, then the homogeneous space $G/B$ is a smooth projective variety. For ...
bernardorim's user avatar
2 votes
1 answer
74 views

Line bundles which are fiber-wise algebraically trivial.

$\DeclareMathOperator{\Pic}{Pic}$ Let $f: M \to B$ be a smooth morphism of smooth varieties over $\mathbb C$ such that the natural map $\mathcal O_B \to f_* \mathcal O_M$ is an isomorphism (i.e. $f$ ...
red_trumpet's user avatar
  • 8,605
0 votes
1 answer
63 views

Connections and curvature on line bundles

I'm trying to understand some basic definitions in relation to line bundles (following Woodhouse's 'Geometric Quantization'). Let $V\rightarrow M$ be a vector bundle (in particular, a complex line ...
A.D.'s user avatar
  • 97
1 vote
0 answers
48 views

Is the tensor product between an ample and free divisor with the power of the ideal sheaf of a point globally generated?

Let $X$ be a nonsingular projective variety over an algebraically closed field of characteristic $0,$ let $H$ be an ample line bundle that is generated by its global sections. If $x$ is any closed ...
MaryMoon's user avatar
0 votes
1 answer
69 views

Identifying sections when computing cohomology groups $H^i(\mathbb P^2,\mathcal O(d))$ via Cech cohomology

I want to compute the dimension and find generators of the cohomology groups $H^i(\mathbb P^2,\mathcal O(d))$. To do so I want to use Cech cohomology and compute directly, or at least try to. So we ...
raisinsec's user avatar
  • 439
0 votes
0 answers
32 views

Exterior Power of a Tensor Product in case one is a line bundle

I was reading this thread Exterior power of a tensor product and I found that the result cited in first answer is very useful to me, but I couldn't prove it myself and what's said there is not enough ...
Emanuele Ronda's user avatar
0 votes
0 answers
87 views

Is there a name for the following important group of line bundles with rational sections?

Is there a name for the following important group of line bundles with rational sections, given by Vakil in FOAG 15.4.3, page 436? 15.4.3. The important group of "line bundles with rational ...
onRiv's user avatar
  • 1,294
1 vote
0 answers
37 views

Why does this curve over $\mathbb{R}$ have no odd degree line bundles?

This is in Moonen's textbook on Abelian Varieties (page 221). My confusion I believe is that I do not understand how to think about $\mathbb{R}$-varieties too well (having only ever worked over $\...
Shrugs's user avatar
  • 1,458
-2 votes
1 answer
289 views

Triviality of line bundles

Let $L$ be a smooth line bundle over $\mathbb{R}$ or $\mathbb{C}$ on a manifold $M$ such that $L^{\otimes m}$ is trivial for some $m \ge 1$. Can we find a smooth $m$-fold covering $(\tilde M,\pi)$ of $...
Summer's user avatar
  • 6,893
3 votes
0 answers
58 views

Parametrizing unstable extensions of line bundles

Suppose $L$ and $M$ are holomorphic line bundles over the same compact Riemann surface $X$. I want to study which extensions of $M$ by $L$, i.e., vector bundles $E$ in a short exact sequence of the ...
Gabriel Martinho's user avatar
0 votes
0 answers
24 views

Criterion of very ampleness of $L^{\otimes 2}$ where $L$ is ample line bundle on abelian variety.

Let $A$ be abelian variety, and $L$ be an ample line bundle on $A$. Then by Lefschetz's theorem $L^{\otimes 3}$ is very ample. I want to know good criterion if $L^{\otimes 2}$ is very ample.
Yos's user avatar
  • 1,924
1 vote
1 answer
136 views

Does the first Chern class generate $H^2(P_n\mathbb{C};\mathbb{Z})$?

Let $P_n\mathbb{C}$ denote the $n$-dimensional complex projective space. We define Chern classes via the Chern-Weil theory, and then I already proved that the first Chern class $-c_1(L)$ for the ...
s.h's user avatar
  • 454
2 votes
0 answers
60 views

When determinant bundle is very ample

For a vector bundle $V$ on a projective variety $X$, let $\Bbb P(V) $ be the projective bundle of hyperplanes. Call $V$ a very ample if $\mathcal O_{\Bbb P(V)}(1)$ is very ample on $\Bbb P(V)$. Let $...
Conjecture's user avatar
  • 3,118
1 vote
0 answers
89 views

Wikipedia wrong on modular forms & line bundles?

The current (25/03/2023 12:30 UTC) version of the Wikipedia article on modular forms has a section "As sections of a line bundle" where it claims the following: I think there are two ...
Johann Birnick's user avatar
3 votes
1 answer
105 views

Show that divisor in $\mathbb P^2\times \mathbb P^2$ is very ample.

Let $H$ denote a quadric hypersurface in $\mathbb P^2\times \mathbb P^2$. In the Chow ring of $\mathbb P^2\times \mathbb P^2$, we have $H\equiv 2H_1+2H_2$, where $H_1,H_2$ are the classes of linear ...
Cellardoor's user avatar
6 votes
1 answer
181 views

Geometric interpretation of the isomorphism $\mathcal{N}_{Y/X} \cong \mathcal{O}_X(Y) \vert_Y$

Let $X$ be a smooth variety / manifold over $\mathbb{C}$ of dimension $n$ and suppose that $Y \subset X$ is a smooth $n-1$-dimensional subvariety. The normal bundle $\mathcal{N}_{Y/X}$ comes from ...
user267839's user avatar
  • 7,315
2 votes
1 answer
113 views

Principal bundle (or torsor) for a diagonalizable group over a torus

Let $T$ be an algebraic torus and $G$ a diagonalizable group; both are over an algebraically closed field $k$ of characteristic $0$ (take $k=\mathbb C$, if you like). I am trying to understand ...
Dave's user avatar
  • 13.6k
1 vote
0 answers
49 views

Does the dual of a line bundle with no sections have a section? [duplicate]

Let $L \to X$ be a holomorphic line bundle over a compact complex manifold. Suppose $L$ is non-trivial and has no non-trivial sections. Let me ask the following (hopefully not entirely trivial) ...
YoungResearcher's user avatar
2 votes
0 answers
150 views

Very ample if and only if algebra generated in degree 1?

Let $f: X \to S$ be a quasi-compact morphism of schemes and $\mathcal{L}$ an invertible $\mathcal{O}_X$-module. Is it true that $\mathcal{L}$ is $f$-very ample if and only if $\mathcal{L}$ is $f$-...
SeparatedScheme's user avatar
0 votes
0 answers
80 views

A connected compact manifold has a unique orientable line bundle on it

How to show that if $M$ is a connected compact manifold there is upto isomorphism a unique orientable line bundle on it and that the line bundle is trivial iff $M$ is orientable?
aritracb's user avatar
  • 735
1 vote
1 answer
160 views

Is the Picard group of a smooth manifold the same as the Picard group of the associated ringed space?

$\def\sO{\mathcal{O}} \def\Pic{\operatorname{Pic}}$Given a ringed space $(X,\sO_X)$, the Picard group of $(X,\sO_X)$ is defined to be the class of isomorphism classes of invertible sheaves of $\sO_X$-...
Elías Guisado Villalgordo's user avatar
3 votes
1 answer
172 views

Is the canonical bundle of a projective complex surface an ample bundle if its self-intersection is positive?

Let $X$ be a projective complex surface ($\dim_\mathbb C X = 2$). If $(K_X, K_X) > 0$, is it true that $K_X$ is an ample line bundle? I am trying to the affirmative answer using Nakai's criterion, ...
rosecabbage's user avatar
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